
Note that taking \(p=2\)  induces a degenerate kernel norm \({- }^2_{^*} = |[] - []|^2\)  which only compares the means. We used \(p=1.5\)  in Figure REF .
We take \(\)  to be a regular discretization of a rotated square in \(^2\) , endowed with the Euclidean distance and uniform measure.
We consider a discrete mm-space parameterized by its support \(x=(x_i)\in (^2)^N\) , i.e. which reads \((x)=(\lbrace x_i\rbrace , {.}_2, (x))\)  where \((x) = \tfrac{1}{N}\sum _i_{x_i}\) .
We perform a gradient flow to minimize \(x\mapsto _{\infty }((x),)\)  and \(x\mapsto _{\infty }((x),)\)  w.r.t to the positions of the samples.
It is similar to the gradient flow experiment of [1]} shown above (Figure REF ), with the difference that the learning rate on the particles' masses is 0 to keep them constant.
The support of \((x_0)\)  is initialized at random on the unit square.
The results are displayed in Figure REF .
We see that the output of the flow for \(_{\infty }\)  seems close to be a square, up to an isometry of \(\)  (rotation and translation).
By comparison, the output of the gradient flow for \(_{\infty }\)  converges to a distribution supported on a disk, and thus ignores the structure of \(\) .
This experiment highlights the bias of entropic regularization and the relevance of correcting it.
It motivates as a future research direction the study of \(_\epsilon \)  for \(\epsilon >0\) .
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